Effect of soluble surfactants on the stability of stratified flows through soft-gel-coated walls

The effect of a soluble surfactant on the linear stability of layered two-phase Poiseuille flows through soft-gel-coated parallel walls is studied in this paper. The focus is on determining the effect of the elastohydrodynamic coupling between the fluids and the soft-gel layers on the various flow instabilities. The fluids are assumed Newtonian and incompressible, while the soft gels are modeled as linear viscoelastic solids. The effect of a soluble surfactant on the different instabilities is specifically investigated. The soft-gel-coated plates are maintained at two different solute concentrations. The dynamics of the soluble surfactant in the fluids is captured using a species transport equation. A linear stability analysis is carried out to identify different instabilities in the system. The linearized governing equations are solved numerically using a Chebyshev spectral Collocation technique. The effect of deformability of the soft gels on three distinct instability modes, (a) a liquid-liquid long-wave mode, (b) a liquid-liquid short-wave mode, and (c) a liquid-liquid Marangoni short-wave mode, is analyzed. An analytical expression for the growth rate is obtained in the long-wave length limit using an asymptotic analysis. From the long-wave analysis a stability map is obtained, in which dominant effects in different regions are identified. The Marangoni stresses can either stabilize or destabilize the interfacial instability depending on the direction of mass transfer. They have a predominantly stabilizing effect on the interfacial instability when the mass transfer is from the more viscous broader fluid to the less viscous thinner fluid. Placing a gel closer to the more viscous fluid has a stabilizing effect on this instability. The Marangoni stresses and soft-gel layers can have opposing effects on the stability of the long-wave mode. The dominant of these two opposing effects is determined by the prevailing parameters. Insights into the dominant physical causes of different instabilities are presented.

Figure 1

Schematic of the flow configuration under study. At the base state, the liquid-liquid interface is at $y=0$, the top gel-liquid interface is at $y=1$, and the bottom gel-liquid interface is at $y=-n_{21}$. The perturbed liquid-liquid interface is given by $y=f\left(x,t\right)$, the perturbed top and bottom gel-liquid interfaces are at $y=1+g\left(x,t\right)$ and $y=-n_{21}+h\left(x,t\right)$, respectively. The top and bottom gel-liquid interfaces are maintained at a solute concentration of $C_{10}$ and $C_{20}$. The piecewise linear base concentration profile and the parabolic velocity profile in the base state for the two fluids are depicted.

Figure 2

The growth rates obtained from the long-wave analysis and numerical simulations using Chebyshev collocation technique for $\mathrm{Ma}=1000$ and 3000. The arrow points in the direction of increasing Ma. Other parameters: $n_{21}=0.5$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\mathrm{W}{\mathrm{i}}_3=0.1$, $\mathrm{W}{\mathrm{i}}_4=0.1$, $\mathrm{C}{\mathrm{a}}_{21}=10$, $\mathrm{C}{\mathrm{a}}_{31}={10}^{-3}$, $\mathrm{C}{\mathrm{a}}_{41}={10}^{-3}$, $K=0.5$, $D_{21}=0.5$, $\gamma =2.2$, $\mathrm{Pe}=2000$, ${\mathrm{Re}}_1=0$.

Figure 3

Division of parameter space into different regions based on the sign of the coefficients ${\alpha }_{\mathrm{Ma}}$, ${\alpha }_{\mathrm{Wi},3}$, and ${\alpha }_{\mathrm{Wi},4}$. Here the direction of the mass transfer is from the bottom to the top gel-liquid interface $\gamma >1/\phantom{1K}K$. Other parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{31}={\mu }_{41}=300$, $D_{21}=0.5$, $K=0.5$, $\gamma =2.2$.

Figure 4

Counteracting effects of Marangoni stresses and soft-gel layers on the stability of the LW mode in regions 1 and 2 in ${\mu }_{21}-n_{21}$ parameter space are shown. Here the direction of the mass transfer is from the bottom to the top gel-liquid interface $\gamma >1/\phantom{1K}K$. Other parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{31}={\mu }_{41}=300$, $D_{21}=0.5$, $K=0.5$, $\gamma =2.2$.

Figure 5

Critical surfaces showing the dominant effect based on the magnitude of coefficients ${\alpha }_{\mathrm{Ma}}$ and ${\alpha }_{\mathrm{Wi},3}$. The arrow points in the direction of increasing fluid two-layer thicknesses. Other parameters: $n_{31}=2$, $n_{41}=3$, ${\mu }_{31}={\mu }_{41}=300$, $D_{21}=0.5$, $K=0.5$, $\gamma =2.2$.

Figure 6

Division of parameter space into different regions based on the sign of the coefficients ${\alpha }_2$, ${\alpha }_3$, and ${\alpha }_4$. Here the direction of the mass transfer is from the top gel-liquid interface to the bottom gel-liquid interface $\gamma <1/\phantom{1K}K$. Other parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{31}={\mu }_{41}=300$, $D_{21}=0.5$, $K=0.5$, $\gamma =0.5$.

Figure 7

Marangoni stresses and soft-gel layers have opposing effects on the stability of the LW mode in regions 3 and 4 in ${\mu }_{21}-n_{21}$ parameter space. Here the direction of the mass transfer is from the top gel-liquid interface to the bottom gel-liquid interface as $\gamma <1/\phantom{1K}K$. Other parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{31}={\mu }_{41}=300$, $D_{21}=0.5$, $K=0.5$, $\gamma =0.5$.

Figure 8

Dispersion curves corresponding to region 4 in Fig. 4, showing that for low Ma growth rates are negative for some wave number $k$ and for higher Ma growth rates are positive for all $k$ (the arrow points in the direction of increasing Ma). This indicates the stabilizing effect of the soluble surfactant. Other parameters: $n_{21}=0.5$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =2.2$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, $\mathrm{C}{\mathrm{a}}_{21}=100$, $\mathrm{C}{\mathrm{a}}_{31}={10}^{-3}$, $\mathrm{C}{\mathrm{a}}_{41}={10}^{-3}$, $\mathrm{W}{\mathrm{i}}_4=1$, $\mathrm{W}{\mathrm{i}}_3=0.1$, $\mathrm{Ma}=1000$, 3000, 5000, and 10 000.

Figure 9

Dispersion curves corresponding to region 4 in Fig. 4, showing that for low ${\mathrm{Wi}}_3$ growth rates are negative for all $k$ and for higher ${\mathrm{Wi}}_3$ growth rates are positive for some $k$ (the arrow points in the direction of increasing ${\mathrm{Wi}}_3$). This indicates the destabilizing effect of the bottom soft-gel layer on the LW mode. Other parameters: $n_{21}=0.5$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =2.2$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1,0.5,1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=1000$.

Figure 10

Eigenfunctions for the LW mode at $x=\frac{2\pi }{k}$ and $\frac{3\pi }{k}$ which are out of phase. Other parameters: $n_{21}=3$, $n_{31}=2$, $n_{41}=4$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =2.2$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=1$, $\mathrm{Ma}=3000$.

Figure 11

Tangential stresses at the top gel-liquid interface and the interfacial velocity at the liquid-liquid interface are in phase. Other parameters: $n_{21}=3$, $n_{31}=2$, $n_{41}=4$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =2.2$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=1$, $\mathrm{Ma}=3000$.

Figure 12

Variation of normalized tangential stresses and the relative velocity of fluids at the liquid-liquid interface along the axial direction. Other parameters: $n_{21}=0.5$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =2.2$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=1000$.

Figure 13

Dispersion curves corresponding to region 3 in Fig. 7. With increase in Ma the magnitude of the growth rate increases (the arrow points in the direction of increasing Ma). This confirms the destabilizing effect of the soluble surfactant on the stability of the LW mode. Here the direction of mass transfer is from the top to bottom gel-liquid interface. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=3000,5000$, and 10 000.

Figure 14

Streamline plots for the LW instability mode. The perturbed interface is shown as a dashed line. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=3000,5000$, and 10 000.

Figure 15

Perturbed axial velocity profiles plotted along the transverse direction at $x=2\pi /k$. The perturbed axial velocity is maximum at the liquid-liquid interface. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=10000$.

Figure 16

Variation of normalized interfacial deformation and normalized first Marangoni stresses along the liquid-liquid interface. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=10000$.

Figure 17

Dispersion curves showing the stabilization of the MSW mode due to deformable nature of the soft-gel layers for ${\mu }_{21}>n_{21}^2$. The magnitudes of the growth rates decrease with increase in ${\mathrm{Wi}}_4$ from 0.1 to 1 (the arrow points in the direction of increasing ${\mathrm{Wi}}_4$). Other parameters: $n_{21}=0.6$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=13000$.

Figure 18

Variation of normalized viscous normal stresses along the interface, plotted along with the normalized interfacial velocity. Other parameters: $n_{21}=0.6$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=13000$.

Figure 19

Streamline contour plot for the MSW instability mode. The perturbed interface is shown by the dashed line. Other parameters: $n_{21}=0.6$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=13000$.

Figure 20

Perturbed axial velocity profiles in the fluids at the liquid-liquid interface plotted at $x=2\pi /k$. The disturbance axial component of velocity is maximum at the liquid-liquid interface. Other parameters: $n_{21}=0.6$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=13000$.

Figure 21

Transition of the LW mode to the MSW mode for $n_{21}=1$ and 0.6. Other parameters: $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=0.59$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=10$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=13000$.

Figure 22

Effect of Marangoni number on the SW mode and MSW mode (the arrow points in the direction of increasing Ma). SW and MSW modes are unstable. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=15000,20000$.

Figure 23

The stabilization of the MSW mode with increase in ${\mathrm{Wi}}_3=1$ and 10 and for ${\mu }_{21}>n_{21}^2$ (the arrow points in the direction of increasing ${\mathrm{Wi}}_3$). Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=20$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, $\mathrm{Ma}=14000$.

Figure 24

Neutral stability curves when both the SW mode and MSW mode are unstable simultaneously. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$.

Figure 25

Variation of normalized second Marangoni stresses along the interface, plotted along with the normalized interfacial velocity for the SW mode. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=15000$.

Figure 26

Switching between MSW and SW modes. The effect of ${\mathrm{Ca}}_{21}$ on these short-wave instabilities is also shown. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$.

Figure 27

Perturbed concentration contour plots for the SW mode shown in Fig. 22. The contours are denser and the gradients along the axial direction are higher at the liquid-liquid interface. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=15000$.

Figure 28

Vorticity contour plot for the SW instability mode. The perturbed interface is shown as a dashed line. Other parameters: $n_{21}=1$, $n_{31}=2$, $n_{41}=2$, ${\mu }_{21}=1.64$, ${\mu }_{31}={\mu }_{41}=300$, $\gamma =0.5$, $K=0.5$, $D_{21}=0.5$, ${\mathrm{Re}}_1=0$, $\mathrm{Pe}=2000$, ${\mathrm{Ca}}_{21}=100$, ${\mathrm{Ca}}_{31}={10}^{-3}$, ${\mathrm{Ca}}_{41}={10}^{-3}$, ${\mathrm{Wi}}_4=0.1$, ${\mathrm{Wi}}_3=0.1$, $\mathrm{Ma}=15000$.